Revisiting Ueda oscillator: Emergence of chaos at ultra-low driving frequencies

Abstract

The degenerate forced Duffing oscillator, characterized by zero linear stiffness and nonzero cubic stiffness, which is commonly known as the Ueda oscillator, has long been regarded as a canonical benchmark in nonlinear dynamics and chaos theory. In his seminal work, Ueda demonstrated the emergence of chaos at a dimensionless unit forcing frequency. In this study, it is shown that chaotic behaviour also arises at ultra-low forcing frequencies. This phenomenon is investigated by constructing basins of attraction in parameter space using the cardinal number measure, and the presence of chaos is further corroborated by bifurcation diagrams and the largest Lyapunov exponent analysis. The observed effect has potential applications in seismic and vibration engineering, as it indicates that long-period ground motions (e.g. near-fault earthquakes) may trigger chaotic responses and elevated acceleration levels in seismically isolated systems; nonlinear energy harvesting, where it opens the possibility of designing broadband nonlinear harvesters capable of efficiently operating under ultra-low-frequency excitation; and other applications, including nonlinear climate dynamics and ecological systems.